Remark 4.5 The ring \(\mathbf {C}[L,B]\)  is a priori only a subring of the affine ring of
the spectral curve, as is clear from Remark REF .
This is a crucial problem, around which we built our algorithm
BC pair, as stated in the Introduction.
Using the parameter \(k\) ,
Segal and Wilson give an illustration of what can be viewed as a containment
of commutative subalgebras, and the surjective morphisms between the attendant
spectral curves [1]}. In particular, if
\(\mathcal {C}_\mathcal {D}(L)=\mathbf {C}[L,B]\) , the spectral curve is special,
in that it can be embedded in the plane with only one smooth point at
infinity; the noted Klein quartic curve gives a non-example of such a curve [2]}. Of course, in the case of a hyperelliptic curve defined by \(B^2\)  equalling a polynomial in \(L\) , the ring of the affine curve is indeed
\(\mathbf {C}[L,B]\) , unless the curve has singular points and
in that case the ring of the desingularization is larger; examples of this can be constructed by transference, but in order to stay in the Weyl algebra, one has to ensure that after conjugation the ring still has polynomial coefficients.

